On the topological centre of the algebra LUC (G)* for general topological groups

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Publication

We consider the Banach algebra LUC (G)* for a not necessarily locally compact topological group G. Our goal is to characterize the topological centre Zt (LUC (G)*) of LUC (G)*. For locally compact groups G, it is well known that Zt (LUC (G)*) equals the measure algebra M (G). We shall prove that for every second countable (not precompact) group G, we have Zt (LUC (G)*) = M (over(G, ̂)), where over(G, ̂) denotes the completion of G with respect to its right uniform structure (if G is precompact, then Zt (LUC (G)*) = LUC (G)*, of course). In fact, this will follow from our more general result stating that for any separable (or any precompact) group G, we have Zt (LUC (G)*) = Leb (G), where Leb (G) denotes the algebra of uniform measures. The latter result also partially answers a conjecture made by I. Csiszár 35 years ago [I. Csiszár, On the weak* continuity of convolution in a convolution algebra over an arbitrary topological group, Studia Sci. Math. Hungar. 6 (1971) 27-40]. We shall give similar results for the topological centre Λ (GLUC) of the LUC-compactification GLUC of G. In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have Λ (GLUC) = over(G, ̂) (if G is precompact, then Λ (GLUC) = GLUC). Finally, we shall show that every linear (left) LUC (G)*-module map on LUC (G) is automatically continuous whenever G is, e.g., separable and not precompact.